Mathemagics: Onslaught Fetchlands by Garrett Johnson

February 11, 2011

I found this really interesting article this morning regarding the math behind fetchlands. it was originally written on tcgplayer, but since digging through archives can be a tedious task. I thought I would share the information with you here. (aka please don’t sue me)

Mathemagics: Onslaught Fetchlands
Garrett Johnson

In my sophomore year getting my bachelor’s degree, I took class in Linear Algebra from a professor by the name of Arthur Benjamin. Among other things, Professor Benjamin had developed a teaching product for teaching math to your children, entitled Mathemagics, which he hocked on late-night TV on a series of Amazing Discoveries infomercials. Think Hooked on Phonics for math. The problem with Mathemagics, however, was that normal kids weren’t as smart as Professor Benjamin. Benjamin was a really smart guy. Really smart. After he took roll in class on the first day, he turned over the list of students and went down each row of students, reciting each of our names from memory. There was a story running around the school, (who know? It may have been true,) that Professor Benjamin was not welcome in Las Vegas, having counted cards, and calculated odds in his head, with sufficient skill as to consistently walk away from a blackjack table in the black. Normal kids, particularly kids who have trouble with math, didn’t have minds that worked like his.

In many ways, deckbuilding in Magic is an equally vexing process. We add an extra Island here, a fourth copy of Cunning Wish there, and we hope that the change will help us, but it’s very difficult, except in certain degenerate cases, (such as an Intuition for three copies of the same card,) to know with any degree of certainty whether you’ve actually made the right choice. This is because the sample size of our games is so very small. Consider that in a 60-card deck there are roughly 2 trillion opening hands you can draw, and that’s before turn one! Even accounting for the degeneracy built into most high-level decks, in the large number of basic lands and the rule of four for many of the most powerful cards, we can still expect that there are more combinations, simply in the opening hand, than even the most devout play testers are capable of seeing, no matter how many times they test a particular build.

Mathematical calculations, though, offer insights into deckbuilding that playtesting would take far too long to discover. For example, the simplest calculation could tell us that in a deck consisting of 20 land and 40 spells, the average number of lands you draw in your opening hand of 7 cards would be 2.33. An only Slightly more complicated calculation would tell us that the odds of opening with X lands in an opening hand of 7 cards would be given by the hypergeometric distribution function, which we can use in Excel:

However, certain cards, most notably the fetch-lands from Onslaught, alter these probabilities. When a Wooded Foothills or a Flooded Strand is sacked for a basic land, the idea is that not only does it give you a land, but it also thins your deck. The hypergeometric function assumes only one card is drawn from the deck, not two, so it’s not possible to determine the effect of fetch-lands on your future draws. This is where a Monte Carlo simulation comes in. A Monte Carlo simulation is a computer program which, instead of calculating out permutations and combinations, instead just starts with the basic odds of drawing each card and simulates a series of turns, many, many times, leveraging the power of a computer to perform a staggering number of simple calculations quickly.

So the question I asked myself was: Are fetch lands all that effective at thinning? Certainly, they function as excellent color fixers when you’re playing them with both allied colors, but I’ve been seeing decklists that increasingly use them not as color fixers, but as land-thinners, such as in Mark Young’s article at StarCityGames.com on June 5, or half a dozen builds using all 8 in the forums for Sligh. The logic is that instead of playing 20 Mountains in your Sligh deck, for example, you play 12, and include 4 Wooded Foothills and 4 Bloodstained Mire. But is this strategy really all that effective at reducing the number of land draws you experience? It certainly worked with Thawing Glaciers, but that was an era when card advantage was king and was with a card that kept on going and going. It was not unusual to see a control player be forced to discard basic lands because he had two Thawing Glaciers bouncing up and down on the table, clearing his deck of land. The fetch lands are only good for one extra land, calling their effectiveness at thinning a deck into question. The best way to put this question to bed is to try it: Try running the first 20 turns of a game say, 1,000,000 times and check out the averages.

The tests were written in java, using J2SDK version 1.4.1, and assumed either 0, 4, 8, or 12 fetch-lands. The total land count, (true land plus fetch lands,) was always maintained at 20, and a total of 1,000,000 games were simulated. The test ignored land type, and assumed that all fetch lands were capable of summoning any of the true lands. The test tracked the number of true lands drawn, the number of fetch lands drawn, and the number of spells drawn. It was run on a 2.4 GHz Pentium 4.

The first test was to observe the effect of Fetch Lands upon the x-by-x probability. The x-by-x probability is the chance of drawing x lands by turn x, thus allowing you to play a land every turn up until turn x. This is important for almost every deck in the game; Aggressive decks need to curve out their mana during the first 3-4 turns, and the x-by-x odds give control decks a great read on how likely mana-screw is, as well as their odds at certain points of having critical amounts of mana. You want to know why control decks usually pack 24+ lands? Check out the terrible odds at 20 land for 4-by-4, a sweet spot for control decks running cards like Wrath of God, Deep Analysis, and Chastise.

The results here are not surprising. The impact of the fetch lands in the early game is minimal, even in the case of 12 fetch lands, despite there only being 8 “true” lands in the whole deck to be played. The x-by-x odds for turns 9-12 in the 12 fetch land case must necessarily be zero, but the odds in the other cases are so close to zero to make no difference. This is not an entirely bad thing – There are very few decks that would be unhappy having 4 lands on turn 4. We want to be drawing a reasonable number of land in our opening hands.

The second test was far more interesting. Here, we postulate a mono-colored aggro deck, (White Weenie and Sligh come to mind,) using a variable number of fetch-lands, assume that an aggro-deck has no use for any land beyond the first few, and simply looked at the total number of nonland cards drawn throughout each game. We would expect that number to be higher for a deck sporting a number of fetch lands, and the difference would be card draws that result in spells instead of land. For an aggro deck that wants to draw nothing but spells after turn 5, this represents effective card advantage. So what did our simulation tell us?

There’s a lot of data on this chart so let me clarify the legend. The black lines represent the 20/0 case, (20 true lands, 0 fetch lands,) the blue lines the 16/4 case, and the green lines the 8/12 case. The thick solid lines are the average number of spells drawn at the given turn, while the thin dotted lines are the average life paid from pain lands. To make this data easier to read, I’ve removed from the next chart the raw number of spells drawn, and only displayed the difference between the average spells drawn for each case, and the 20/0 control group, and I’ve added the Dead Draw average, the average number of useless fetch lands we’d expect to draw when the deck’s already been exhausted of true land, denoted by dashed lines. I’ve also limited the number of turns we look at to 16, as any Sligh or White Weenie deck still playing on turn 16 has already lost.

This is fairly surprising. The overall impact of fetch lands upon the number of extra spells, (as opposed to the true lands the fetch lands removed from the deck,) drawn is not nearly as high as the number life sacrificed for the effect. While the dead draws over the first 16 turns are not surprisingly negligible, over the first 16 turns we cannot realize, on average, a single extra card from our fetch lands. Even if we propagate this data further, the first card we see in the 4/16 case is not realized until around the 36th turn, and at an average cost of 2.8 life. The first card for 8/12 is realized on the 25th turn, but at a cost of 4.3 life.

Obviously, these numbers are merely averages, but they do tell us that it takes an awful long time for the card advantage to be realized. It’s impossible to determine exactly when the extra would actually be drawn, given the randomness of the system, but we can expect it to be close to the turns noted above. What’s happening is that despite the life being paid immediately, the advantage of extra cards are realize quite a long time later, as the probabilities begin to skew upwards as more cards are being drawn, and as more lands are later being thinned. A simple exchange of 4 life for one card, seen in the 8/12 case, may be palatable to a Sligh deck, but the deferral of that card being drawn dramatically decreases its value, even though you don’t pay all 4 life up front, either. Consider how effective cards that mimic this effect are against aggro decks: Unsummon, Repulse, and Hibernation have all seen significant play as tech against aggro decks because the loss of tempo is so crippling to an aggro deck. This is despite the card neutrality or card disadvantage generated by these cards. Even in the degenerate 12/8 case, (not shown since it’s limited to Type I decks running dual lands,) the extra card drawn on turn 21 is far too late, particularly in Type I.

The focus here has been on deck thinning, and the card advantage generated from it in the context of an aggro, usually mono-colored deck. There are certainly other variables that might warrant the inclusion of the Onslaught fetch-lands, which include, but aren’t necessarily limited to color fixing, graveyard filling, and reshuffling, (after a Brainstorm, for example.) Wooded Foothills fixes colors beautifully in RG beats and many controlling decks such as Psychatog and Wake make excellent use of Polluted Delta or Flooded Strand to fix their mana. Threshold decks can certainly make use of reasonable number of fetch-lands to trade life for graveyard cards on a one-for-one basis.

However, these arguments don’t apply to mono-colored aggro, with the sole exception of Grim Lavamancer in Sligh. Moreover, there is one potentially large reason not to haphazardly play fetch-lands: Stifle. While it’s difficult to gauge a metagame that has not yet developed, the prospect of a first turn Stifle on your Wooded Foothills when you’re playing second is pretty crippling. Stifle is more than landkill in that case. It’s land countering, functionally a turn one Time Walk. Finally, the fetch-lands can be painful, even for Sligh, in the mirror match, when both life totals can get very low, regardless of “who’s the control deck.”

Ultimately, then, I would argue that the data bears out the contention that playing fetch-lands for their thinning effects are a bad idea: Only a suicidally reckless aggro deck can afford 4 life for a card, and those decks can’t afford to wait 20+ turns for it.

One other note: I can’t argue with the people who tell us in articles that they’ve observed the thinning effect of fetch lands except to point out that thinning is an extremely difficult thing to see during a game. Few people count the number of lands they draw, and even if they did, their perceptions would be colored by what they’re trying to do during the game. A land draw’s a lot more palatable with a Deep Analysis and a Roar Of The Wurm in hand or graveyard than when you’ve got a Merfolk Looter and a Basking Rootwalla in hand. And anyone who tells you that they can simply tell they’re drawing fewer land has never heard of a standard deviation.

The Monte Carlo simulation can be a powerful tool in deckbuilding, providing insight into problems that cannot be easily solved analytically. …And its power has only been touched upon in the two simulations I ran. It’s possible to postulate much more complicated situations, such as the odds of drawing a madness enabler and an Arrogant Wurm in a UG madness deck, or the average goldfish kill time in a Sligh deck. They key is posing a very narrow question that a simulation can answer unequivocally. Next week, we’ll be using it to look at Krosan Verge.